Event Begins: Wednesday, November 13, 2024 4:00pm
Location: East Hall
Organized By: Algebraic Geometry Seminar - Department of Mathematics

Joint work with Ailsa Keating (Cambridge). We prove the homological mirror symmetry conjecture of Kontsevich for K3 surfaces in the following form: The Fukaya category of a projective K3 surface is equivalent to the derived category of coherent sheaves on the mirror, which is a K3 surface of Picard rank 19 over the field of formal Laurent series. This builds on prior work of Seidel (who proved the theorem in the case of the quartic surface), Sheridan, Lekili--Ueda, and Ganatra--Pardon--Shende.

I will try to keep prerequisites to a minimum, in particular, I will not assume prior knowledge of the Fukaya category.

Event Begins: Wednesday, November 13, 2024 3:00pm
Location: East Hall
Organized By: Learning Seminar in Algebraic Combinatorics - Department of Mathematics

By what has been shown in previous talks, we have seen that we can show coefficients of the characteristic polynomial of a realizable matroid can be realized via specific computations in the Chow ring of its wonderful compactification. In this talk, we will introduce the notion of Poincare duality algebras, which are graded algebras with a degree function giving an isomorphism from the top degree to the base field that induces a non-degenerate pairing between complementary degrees of the algebra. Furthermore, we will introduce a notion of hard Lefschetz and Hodge-Riemann relations for such algebras. When a Poincare duality algebra satisfies a certain version of these properties, we can show that the log-concavity of its "volume polynomial" is equivalent to the eigenvalues of a symmetric form on the algebra arising from the Hodge-Riemann relations. Because the Hodge-Riemann relations in appropriate degree imply the log-concavity of the coefficients of the characteristic polynomial of the matroid, this framework gives us a program to establish the log-concavity result. Throughout this talk, I will attempt to provide intuition from the case of the Chow rings of smooth projective varieties.

Toni Annala, Marc Hoyois and Ryomei Iwasa
J. Amer. Math. Soc. 38 (), 243-289.
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Gurmeet K. Bakshi, Jyoti Garg and Gabriela Olteanu
Math. Comp. 93 (), 3027-3058.
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Alin Bostan, Xavier Caruso and Julien Roques
Bull. Amer. Math. Soc. 61 (), 609-658.
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An educator's experience teaching math is important, but performance on math-content-certification tests is the best predictor of how well a teacher's students will perform in early algebra, finds a new study by the Regional Educational Laboratory Central at Marzano Research.

New York : Springer, 2005

Basel ; Boston : Birkhauser, 2007

[Place of publication not identified] : American Mathematical Society, 2011.

Boston ; Berlin : Birkhäuser, 2006

New York : Springer, 2005

New York : Springer, 2007

Cham, Switzerland : Springer, 2021.

Singapore : World Scientific Publishing Company, 2022.

Cham : Springer International Publishing : Imprint: Springer, 2018.

Cham, Switzerland : Springer, 2018

Cham : Springer, 2019.

We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.

The $(q, mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, mathbf{Q})$-current algebra $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$ associated with the special linear Lie algebra $mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$-modules.

We derive the non-relativistic $c oinfty$ and ultra-relativistic $c o 0$ limits of the $kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $kappa$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $kappa$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincar'e, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $kappa$-Newtonian and $kappa$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $kappa$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $kappa$, the curvature parameter $eta$ and the speed of light parameter $c$.

Yang-Mills gauge theory on Minkowski space supports a Batalin-Vilkovisky-infinity algebra structure, all whose operations are local. To make this work, the axioms for a BV-infinity algebra are deformed by a quadratic element, here the Minkowski wave operator. This homotopy structure implies BCJ/color-kinematics duality; a cobar construction yields a strict algebraic structure whose Feynman expansion for Yang-Mills tree amplitudes complies with the duality. It comes with a `syntactic kinematic algebra'.

We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, strongly purely infinite C*-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable $KK$-contractible C*-algebras: Two Rokhlin flows on such a C*-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.

We present a systematic derivation of the abelianity conditions for the $q$-deformed $W$-algebras constructed from the elliptic quantum algebra $mathcal{A}_{q,p}(widehat{gl}(N)_{c})$. We identify two sets of conditions on a given critical surface yielding abelianity lines in the moduli space ($p, q, c$). Each line is identified as an intersection of a countable number of critical surfaces obeying diophantine consistency conditions. The corresponding Poisson brackets structures are then computed for which some universal features are described.

Axial algebras are non-associative algebras generated by semisimple idempotents whose adjoint actions obey a fusion law. Axial algebras that are generated by two such idempotents play a crucial role in the theory. We classify all primitive 2-generated axial algebras whose fusion laws have two eigenvalues and all graded primitive 2-generated axial algebras whose fusion laws have three eigenvalues. This represents a significant broadening in our understanding of axial algebras.

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $mathbb{R}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of [Aschieri et al.,Class. Quantum Gravity 23 (2006), 1883], whereby the commutative pointwise product is replaced by the $star$-product determined by a Drinfel'd twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are level sets of the $f^a(x)$, where $f^a(x)=0$ are the polynomial equations solved by the points of $M$, employing twists based on the Lie algebra $Xi_t$ of vector fields that are tangent to all the $M_c$. The twisted Cartan calculus is automatically equivariant under twisted $Xi_t$. If we endow $mathbb{R}^n$ with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted $M$ is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and $star$-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $mathbb{R}^3$ except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $mathbb{R}^3$ and twisted hyperboloids embedded in twisted Minkowski $mathbb{R}^3$ [the latter are twisted (anti-)de Sitter spaces $dS_2,AdS_2$].

In recent years, a large class of nuclear $C^ast$-algebras have been classified, modulo an assumption on the Universal Coefficient Theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising.

We equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the $p$-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel.

Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable.

High-performance implementations of graph algorithms are challenging to implement on new parallel hardware such as GPUs, because of three challenges: (1) difficulty of coming up with graph building blocks, (2) load imbalance on parallel hardware, and (3) graph problems having low arithmetic intensity. To address these challenges, GraphBLAS is an innovative, on-going effort by the graph analytics community to propose building blocks based in sparse linear algebra, which will allow graph algorithms to be expressed in a performant, succinct, composable and portable manner. In this paper, we examine the performance challenges of a linear algebra-based approach to building graph frameworks and describe new design principles for overcoming these bottlenecks. Among the new design principles is exploiting input sparsity, which allows users to write graph algorithms without specifying push and pull direction. Exploiting output sparsity allows users to tell the backend which values of the output in a single vectorized computation they do not want computed. Load-balancing is an important feature for balancing work amongst parallel workers. We describe the important load-balancing features for handling graphs with different characteristics. The design principles described in this paper have been implemented in "GraphBLAST", the first open-source linear algebra-based graph framework on GPU targeting high-performance computing. The results show that on a single GPU, GraphBLAST has on average at least an order of magnitude speedup over previous GraphBLAS implementations SuiteSparse and GBTL, comparable performance to the fastest GPU hardwired primitives and shared-memory graph frameworks Ligra and Gunrock, and better performance than any other GPU graph framework, while offering a simpler and more concise programming model.

Randomized Numerical Linear Algebra (RandNLA) uses randomness to develop improved algorithms for matrix problems that arise in scientific computing, data science, machine learning, etc. Determinantal Point Processes (DPPs), a seemingly unrelated topic in pure and applied mathematics, is a class of stochastic point processes with probability distribution characterized by sub-determinants of a kernel matrix. Recent work has uncovered deep and fruitful connections between DPPs and RandNLA which lead to new guarantees and improved algorithms that are of interest to both areas. We provide an overview of this exciting new line of research, including brief introductions to RandNLA and DPPs, as well as applications of DPPs to classical linear algebra tasks such as least squares regression, low-rank approximation and the Nystr"om method. For example, random sampling with a DPP leads to new kinds of unbiased estimators for least squares, enabling more refined statistical and inferential understanding of these algorithms; a DPP is, in some sense, an optimal randomized algorithm for the Nystr"om method; and a RandNLA technique called leverage score sampling can be derived as the marginal distribution of a DPP. We also discuss recent algorithmic developments, illustrating that, while not quite as efficient as standard RandNLA techniques, DPP-based algorithms are only moderately more expensive.

Yves André
J. Amer. Math. Soc. 33 (2020), 363-380.
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Yifei Zhu
Trans. Amer. Math. Soc. 373 (2020), 3733-3764.
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Burt Totaro
Trans. Amer. Math. Soc. 373 (2020), 3609-3626.
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Boris Adamczewski, Julien Cassaigne and Marion Le Gonidec
Trans. Amer. Math. Soc. 373 (2020), 3085-3115.
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Michiya Mori
Proc. Amer. Math. Soc. 148 (2020), 2477-2485.
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Daniel Carando, Santiago Muro and Daniela M. Vieira
Proc. Amer. Math. Soc. 148 (2020), 2447-2457.
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Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos and Andrea Solotar
Proc. Amer. Math. Soc. 148 (2020), 2421-2432.
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Véronique Bazier-Matte and Pierre-Guy Plamondon
Proc. Amer. Math. Soc. 148 (2020), 2397-2409.
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Javad Asadollahi and Rasool Hafezi
Proc. Amer. Math. Soc. 148 (2020), 2379-2396.
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Joseph Reid
Proc. Amer. Math. Soc. 148 (2020), 2331-2343.
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Claus Sorensen
Represent. Theory 24 (2020), 151-177.
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Vladimir Dragović and Irina Goryuchkina
Bull. Amer. Math. Soc. 57 (2020), 293-299.
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